How to understand this kind of equivalence? When we study algebraic structures there is always the notion of "isomorphism", which intuitively means that two instances of the same algebraic structure are "algebraically indistinguishable". For example, every real vector space of dimension $n < \infty$ is isomorphic to $\mathbb{R}^n$. This kind of equivalence I think I understand.
Here is another slightly different (in my opinion) kind of equivalence I think I understand: The principle of mathematical induction is equivalent to the Well-ordering principle. Up to some details (i.e., intuitively), this basically means that both principles have "the same consequences" in the formal axiomatic system (again, vague but may be completely formalized). In the same vein: Axiom of choice is equivalent to Zorn's lemma. Lots of statements equivalent to the Parallel postulate in Euclidean Geometry, etc.
Here is one kind of equivalence I don't understand.


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*"A group is the same thing as a category with exactly one object and such that every morphism is an isomorphism".

*"Consider $R$ a commutative ring with identity. A $R$-module is the same thing as an abelian group $M$ together with a ring homomorphism $\varphi: R \to End(M)$."

*Suppose $A,B$ commutative rings with identity. A ring $B$ together with a ring homomorphism $\varphi:A \to B$ is the same thing as an $A$-algebra.


The three concepts "defined" above have their "standard definition" as a set with some additional structure.
The above three examples are quite different from the previous examples of equivalence. I "know" what these sentences mean, but I don't understand them.
I can show, for example, that given a category $C$ in which every morphism is an isomorphism and such that $C$ has only one object, the set $Hom(C)$ is a group. Conversely, given a group $G$ I can define a category with only one (formal) object and in which the arrows are the elements of $G$, composition of arrows is multiplication of elements in $G$, this constitutes a perfect category. But this is all the intuition I have about this.
Why are the above equivalences useful?
How can I understand it better?
What is the point of, for example, "defining" a group as a special category? Is it just to show the "expressive power" of our new language? Is this notion of equivalence formalized in some language (category theory, perhaps)?
I've been studying category theory for some weeks now and I don't see the utility of being able to define a group as a category. This makes me a little bit frustrated because it's an exercise in every book on category theory I look at and I don't completely see the point of it.
I know my question is vague, I appreciate any feedback.
Thank you. 
 A: All three numbered items can be formalized as an equivalence of categories which is something that should be covered by any introductory book on category theory.  For instance, number 2 could be stated as:
Let $RMod$ denote the category of groups, and $C$ the category of abelian groups $M$ with a ring homomorphism $R \to \mathrm{End}(M)$ (with morphisms in $C$ being defined by morphisms $M_1 \to M_2$ which make a natural diagram commute).  Then there is a canonical equivalence of categories $RMod \simeq C$.
(The notion of equivalence of categories is the most natural way to generalize the notion of isomorphism to the situation of categories.  In the example above, depending on the exact definitions of functors $RMod \to C$ and $C \to RMod$, the compositions won't necessarily give you literally the same object back - and according to the philosophy of category theory, that's fine: the exact object isn't important, only the isomorphism class.  So, what's really important is that you get an object that's isomorphic to the original object, in a "natural" way - where the definition of "natural" is historically the key insight that made category theory actually start to be interesting and useful.)
As far as the point of the equivalence in #1: for instance, it would allow you to apply any theorem about categories, or about groupoids in particular, and get a theorem about groups.
